Manual model development
1 MVP
You are given a set of data on housing sale prices for the last few years in King County (near Seattle) between May 2014 and May 2015.
We want you to build an explanatory model for the price of housing in King County, i.e. an interpretable model in which the included variables are statistically justifiable.
The variable definitions are:
id - Unique ID for each home sold
date - Date of the home sale
price - Price of each home sold
bedrooms - Number of bedrooms
bathrooms - Number of bathrooms, where .5 accounts for a room with a toilet but no shower
sqft_living - Square footage of the apartments interior living space
sqft_lot - Square footage of the land space
floors - Number of floors
waterfront - A dummy variable for whether the apartment was overlooking the waterfront or not
view - An index from 0 to 4 of how good the view of the property was
condition - An index from 1 to 5 on the condition of the apartment
grade - An index from 1 to 13, where 1-3 falls short of building construction and design, 7 has an average level of construction and design, and 11-13 have a high quality level of construction and design
sqft_above - The square footage of the interior housing space that is above ground level
sqft_basement - The square footage of the interior housing space that is below ground level
yr_built - The year the house was initially built
yr_renovated - The year of the house’s last renovation
zipcode - What zipcode area the house is in
lat - Lattitude
long - Longitude
sqft_living15 - The square footage of interior housing living space for the nearest 15 neighbors
sqft_lot15 - The square footage of the land lots of the nearest 15 neighbors
2 Question 1
Tidy up the data ready for regression:
* You might like to think about removing some or all of `date`, `id`, `sqft_living15`, `sqft_lot15` and `zipcode` (`lat` and `long` provide a better measure of location in any event).
* Have a think about how to treat `waterfront`. Should we convert its type?
* We converted `yr_renovated` into a `renovated` logical variable, indicating whether the property had ever been renovated. You may wish to do the same.
* Have a think about how to treat `condition` and `grade`? Are they interval or categorical ordinal data types?library(tidyverse)## Warning: replacing previous import 'vctrs::data_frame' by
## 'tibble::data_frame' when loading 'dplyr'## Warning: package 'ggplot2' was built under R version 3.6.2## Warning: package 'tibble' was built under R version 3.6.2## Warning: package 'tidyr' was built under R version 3.6.2## Warning: package 'purrr' was built under R version 3.6.2## Warning: package 'dplyr' was built under R version 3.6.2library(GGally)## Warning: package 'GGally' was built under R version 3.6.2library(modelr)houses <- read_csv("data/kc_house_data.csv")## Parsed with column specification:
## cols(
## .default = col_double(),
## id = col_character(),
## date = col_datetime(format = "")
## )## See spec(...) for full column specifications.glimpse(houses)## Rows: 21,613
## Columns: 21
## $ id <chr> "7129300520", "6414100192", "5631500400", "2487200…
## $ date <dttm> 2014-10-13, 2014-12-09, 2015-02-25, 2014-12-09, 2…
## $ price <dbl> 221900, 538000, 180000, 604000, 510000, 1225000, 2…
## $ bedrooms <dbl> 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 5, 4, 3,…
## $ bathrooms <dbl> 1.00, 2.25, 1.00, 3.00, 2.00, 4.50, 2.25, 1.50, 1.…
## $ sqft_living <dbl> 1180, 2570, 770, 1960, 1680, 5420, 1715, 1060, 178…
## $ sqft_lot <dbl> 5650, 7242, 10000, 5000, 8080, 101930, 6819, 9711,…
## $ floors <dbl> 1.0, 2.0, 1.0, 1.0, 1.0, 1.0, 2.0, 1.0, 1.0, 2.0, …
## $ waterfront <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ view <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ condition <dbl> 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3,…
## $ grade <dbl> 7, 7, 6, 7, 8, 11, 7, 7, 7, 7, 8, 7, 7, 7, 7, 9, 7…
## $ sqft_above <dbl> 1180, 2170, 770, 1050, 1680, 3890, 1715, 1060, 105…
## $ sqft_basement <dbl> 0, 400, 0, 910, 0, 1530, 0, 0, 730, 0, 1700, 300, …
## $ yr_built <dbl> 1955, 1951, 1933, 1965, 1987, 2001, 1995, 1963, 19…
## $ yr_renovated <dbl> 0, 1991, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ zipcode <dbl> 98178, 98125, 98028, 98136, 98074, 98053, 98003, 9…
## $ lat <dbl> 47.5112, 47.7210, 47.7379, 47.5208, 47.6168, 47.65…
## $ long <dbl> -122.257, -122.319, -122.233, -122.393, -122.045, …
## $ sqft_living15 <dbl> 1340, 1690, 2720, 1360, 1800, 4760, 2238, 1650, 17…
## $ sqft_lot15 <dbl> 5650, 7639, 8062, 5000, 7503, 101930, 6819, 9711, …# tidy up data. In particular treat condition and grade as factor, as they are
# ordinal categorical
houses_tidy <- houses %>%
select(-c("id", "date", "sqft_living15", "sqft_lot15", "zipcode")) %>%
mutate(waterfront = as.logical(waterfront)) %>%
mutate(renovated = yr_renovated != 0) %>%
select(-"yr_renovated") %>%
mutate(condition = as_factor(condition)) %>%
mutate(grade = as_factor(grade))
glimpse(houses_tidy)## Rows: 21,613
## Columns: 16
## $ price <dbl> 221900, 538000, 180000, 604000, 510000, 1225000, 2…
## $ bedrooms <dbl> 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 5, 4, 3,…
## $ bathrooms <dbl> 1.00, 2.25, 1.00, 3.00, 2.00, 4.50, 2.25, 1.50, 1.…
## $ sqft_living <dbl> 1180, 2570, 770, 1960, 1680, 5420, 1715, 1060, 178…
## $ sqft_lot <dbl> 5650, 7242, 10000, 5000, 8080, 101930, 6819, 9711,…
## $ floors <dbl> 1.0, 2.0, 1.0, 1.0, 1.0, 1.0, 2.0, 1.0, 1.0, 2.0, …
## $ waterfront <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, F…
## $ view <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ condition <fct> 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3,…
## $ grade <fct> 7, 7, 6, 7, 8, 11, 7, 7, 7, 7, 8, 7, 7, 7, 7, 9, 7…
## $ sqft_above <dbl> 1180, 2170, 770, 1050, 1680, 3890, 1715, 1060, 105…
## $ sqft_basement <dbl> 0, 400, 0, 910, 0, 1530, 0, 0, 730, 0, 1700, 300, …
## $ yr_built <dbl> 1955, 1951, 1933, 1965, 1987, 2001, 1995, 1963, 19…
## $ lat <dbl> 47.5112, 47.7210, 47.7379, 47.5208, 47.6168, 47.65…
## $ long <dbl> -122.257, -122.319, -122.233, -122.393, -122.045, …
## $ renovated <lgl> FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FA…3 Question 2
Check for aliased variables using the alias() function (this takes in a formula object and a data set). [Hint - formula price ~ . says ‘price varying with all predictors’, this is a suitable input to alias()]. Remove variables that lead to an alias. Check the ‘Elements of multiple regression’ lesson for a dropdown containing further information on finding aliased variables in a dataset.
Solution
# Alias is useful to check if we have aliased variables, i.e. one or more
# variables that can be computed from other variables
alias(price ~ ., data = houses_tidy)## Model :
## price ~ bedrooms + bathrooms + sqft_living + sqft_lot + floors +
## waterfront + view + condition + grade + sqft_above + sqft_basement +
## yr_built + lat + long + renovated
##
## Complete :
## (Intercept) bedrooms bathrooms sqft_living sqft_lot floors
## sqft_basement 0 0 0 1 0 0
## waterfrontTRUE view condition2 condition3 condition4
## sqft_basement 0 0 0 0 0
## condition5 grade3 grade4 grade5 grade6 grade7 grade8 grade9
## sqft_basement 0 0 0 0 0 0 0 0
## grade10 grade11 grade12 grade13 sqft_above yr_built lat long
## sqft_basement 0 0 0 0 -1 0 0 0
## renovatedTRUE
## sqft_basement 0# seems that sqft_basement can be computed from sqft_living - sqft_above.
# let's drop sqft_living leaving just the two contributions sqft_basement and
# sqft_above
houses_tidy <- houses_tidy %>%
select(-"sqft_living")
glimpse(houses_tidy)## Rows: 21,613
## Columns: 15
## $ price <dbl> 221900, 538000, 180000, 604000, 510000, 1225000, 2…
## $ bedrooms <dbl> 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 5, 4, 3,…
## $ bathrooms <dbl> 1.00, 2.25, 1.00, 3.00, 2.00, 4.50, 2.25, 1.50, 1.…
## $ sqft_lot <dbl> 5650, 7242, 10000, 5000, 8080, 101930, 6819, 9711,…
## $ floors <dbl> 1.0, 2.0, 1.0, 1.0, 1.0, 1.0, 2.0, 1.0, 1.0, 2.0, …
## $ waterfront <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, F…
## $ view <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ condition <fct> 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3,…
## $ grade <fct> 7, 7, 6, 7, 8, 11, 7, 7, 7, 7, 8, 7, 7, 7, 7, 9, 7…
## $ sqft_above <dbl> 1180, 2170, 770, 1050, 1680, 3890, 1715, 1060, 105…
## $ sqft_basement <dbl> 0, 400, 0, 910, 0, 1530, 0, 0, 730, 0, 1700, 300, …
## $ yr_built <dbl> 1955, 1951, 1933, 1965, 1987, 2001, 1995, 1963, 19…
## $ lat <dbl> 47.5112, 47.7210, 47.7379, 47.5208, 47.6168, 47.65…
## $ long <dbl> -122.257, -122.319, -122.233, -122.393, -122.045, …
## $ renovated <lgl> FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FA…4 Question 3
Systematically build a regression model containing up to four main effects (remember, a main effect is just a single predictor with coefficient), testing the regression diagnostics as you go
* splitting datasets into numeric and non-numeric columns might help `ggpairs()` run in manageable time, although you will need to add either a `price` or `resid` column to the non-numeric dataframe in order to see its correlations with the non-numeric predictors. You can do it in the following way:houses_tidy_numeric <- houses_tidy %>%
select_if(is.numeric)
houses_tidy_nonnumeric <- houses_tidy %>%
select_if(function(x) !is.numeric(x))
houses_tidy_nonnumeric$price <- houses_tidy$priceSolution
ggpairs(houses_tidy_numeric)
ggpairs(houses_tidy_nonnumeric)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
You can also subset your ggpairs plot doing the following:
ggpairs(houses_tidy_numeric, columns = c(1, 2, 3, 4))
and the same in subsequent rounds of predictor selection with the resid column.
Remember, if you are not sure whether including a categorical predictor is statistically justified, run an anova() test passing in the models with- and without the categorical predictor and check the p-value of the test.
houses_tidy_numeric <- houses_tidy %>%
select_if(is.numeric)
houses_tidy_nonnumeric <- houses_tidy %>%
select_if(function(x) !is.numeric(x))
houses_tidy_nonnumeric$price <- houses_tidy$price
ggpairs(houses_tidy_numeric)
ggpairs(houses_tidy_nonnumeric)
Correlation of sqft_above with price looks pretty promising, but split of price by grade and waterfront also look decent.
houses_tidy %>%
ggplot(aes(x = grade, y = price)) +
geom_boxplot()
houses_tidy %>%
ggplot(aes(x = waterfront, y = price)) +
geom_boxplot()
Start building the models:
# model with price and sqft above
mod1a <- lm(price ~ sqft_above, data = houses_tidy)
summary(mod1a)##
## Call:
## lm(formula = price ~ sqft_above, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -913132 -165624 -41468 109327 5339232
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 59953.2 4729.8 12.68 <2e-16 ***
## sqft_above 268.5 2.4 111.87 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 292200 on 21611 degrees of freedom
## Multiple R-squared: 0.3667, Adjusted R-squared: 0.3667
## F-statistic: 1.251e+04 on 1 and 21611 DF, p-value: < 2.2e-16mod1b <- lm(price ~ grade, data = houses_tidy)
summary(mod1b)##
## Call:
## lm(formula = price ~ grade, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1929615 -135853 -35090 89080 5565658
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 142000 254499 0.558 0.576878
## grade3 63667 293870 0.217 0.828484
## grade4 72381 258849 0.280 0.779767
## grade5 106524 255024 0.418 0.676169
## grade6 159920 254561 0.628 0.529868
## grade7 260590 254513 1.024 0.305904
## grade8 400853 254520 1.575 0.115285
## grade9 631513 254547 2.481 0.013112 *
## grade10 929771 254611 3.652 0.000261 ***
## grade11 1354842 254817 5.317 1.07e-07 ***
## grade12 2049222 255909 8.008 1.23e-15 ***
## grade13 3567615 264106 13.508 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 254500 on 21601 degrees of freedom
## Multiple R-squared: 0.5197, Adjusted R-squared: 0.5195
## F-statistic: 2125 on 11 and 21601 DF, p-value: < 2.2e-16mod1c <- lm(price ~ waterfront, data = houses_tidy)
summary(mod1c)##
## Call:
## lm(formula = price ~ waterfront, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1376876 -211564 -81564 108436 7168436
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 531564 2416 220.00 <2e-16 ***
## waterfrontTRUE 1130312 27822 40.63 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 353900 on 21611 degrees of freedom
## Multiple R-squared: 0.07095, Adjusted R-squared: 0.07091
## F-statistic: 1650 on 1 and 21611 DF, p-value: < 2.2e-16Grade looks the most promising, but some of the grade level coefficients are insignificant.
From the F-test at the bottom of the regression output tests against the null model (i.e. intercept only). But, if we want, we can replicate this using a separate anova. For this, our null model would be to regress price on intercept only.
You can do this in the following way:
null_model <- lm(price ~ 1, data = houses_tidy)
grade_model <- lm(price ~ grade, data = houses_tidy)
anova(null_model, grade_model)# grade is significant, let's keep it. Now plot diagnostics
par(mfrow = c(2, 2))
plot(mod1b)## Warning: not plotting observations with leverage one:
## 19453
## Warning: not plotting observations with leverage one:
## 19453
houses_resid <- houses_tidy %>%
add_residuals(mod1b) %>%
select(-c("price", "grade"))
houses_resid_numeric <- houses_resid %>%
select_if(is.numeric)
houses_resid_nonnumeric <- houses_resid %>%
select_if(function(x) !is.numeric(x))
houses_resid_nonnumeric$resid <- houses_resid$residggpairs(houses_resid_numeric)
ggpairs(houses_resid_nonnumeric)
Our predictor lat has highest correlation with residuals, but, again, waterfront still looks pretty promising. Try both…
mod2a <- lm(price ~ grade + lat, data = houses_tidy)
summary(mod2a)##
## Call:
## lm(formula = price ~ grade + lat, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1863783 -112379 -28181 67454 5533910
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -29949769 610666 -49.044 < 2e-16 ***
## grade3 187923 276126 0.681 0.496151
## grade4 95367 243212 0.392 0.694978
## grade5 126354 239618 0.527 0.597980
## grade6 158453 239183 0.662 0.507673
## grade7 246275 239138 1.030 0.303093
## grade8 378635 239144 1.583 0.113369
## grade9 603010 239170 2.521 0.011701 *
## grade10 891752 239230 3.728 0.000194 ***
## grade11 1311124 239425 5.476 4.4e-08 ***
## grade12 2007093 240450 8.347 < 2e-16 ***
## grade13 3502099 248154 14.113 < 2e-16 ***
## lat 633100 11822 53.554 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 239100 on 21600 degrees of freedom
## Multiple R-squared: 0.576, Adjusted R-squared: 0.5758
## F-statistic: 2445 on 12 and 21600 DF, p-value: < 2.2e-16mod2b <- lm(price ~ grade + waterfront, data = houses_tidy)
summary(mod2b)##
## Call:
## lm(formula = price ~ grade + waterfront, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1929615 -132969 -32393 91481 4778940
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 142000 244366 0.581 0.5612
## grade3 63667 282169 0.226 0.8215
## grade4 72381 248543 0.291 0.7709
## grade5 92834 244870 0.379 0.7046
## grade6 155043 244426 0.634 0.5259
## grade7 258469 244379 1.058 0.2902
## grade8 395393 244386 1.618 0.1057
## grade9 623595 244413 2.551 0.0107 *
## grade10 909321 244474 3.719 0.0002 ***
## grade11 1313326 244674 5.368 8.06e-08 ***
## grade12 1947993 245731 7.927 2.35e-15 ***
## grade13 3567615 253590 14.068 < 2e-16 ***
## waterfrontTRUE 828234 19363 42.773 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 244400 on 21600 degrees of freedom
## Multiple R-squared: 0.5572, Adjusted R-squared: 0.557
## F-statistic: 2265 on 12 and 21600 DF, p-value: < 2.2e-16# lat is significant and higher r^2, let's keep model2a
par(mfrow = c(2, 2))
plot(mod2a)## Warning: not plotting observations with leverage one:
## 19453
## Warning: not plotting observations with leverage one:
## 19453
houses_resid <- houses_tidy %>%
add_residuals(mod2a) %>%
select(-c("price", "grade", "lat"))
houses_resid_numeric <- houses_resid %>%
select_if(is.numeric)
houses_resid_nonnumeric <- houses_resid %>%
select_if(function(x) !is.numeric(x))
houses_resid_nonnumeric$resid <- houses_resid$residggpairs(houses_resid_numeric)
ggpairs(houses_resid_nonnumeric)
Now view has strongest correlation with residuals, but also compare against model with waterfront.
mod3a <- lm(price ~ grade + lat + view, data = houses_tidy)
summary(mod3a)##
## Call:
## lm(formula = price ~ grade + lat + view, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1665265 -105866 -21393 69623 5429667
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -30514024 576750 -52.907 < 2e-16 ***
## grade3 190253 260743 0.730 0.465607
## grade4 81058 229663 0.353 0.724133
## grade5 112154 226269 0.496 0.620134
## grade6 148515 225858 0.658 0.510827
## grade7 235345 225815 1.042 0.297328
## grade8 351873 225822 1.558 0.119203
## grade9 556584 225848 2.464 0.013731 *
## grade10 821118 225907 3.635 0.000279 ***
## grade11 1200228 226097 5.308 1.12e-07 ***
## grade12 1832950 227080 8.072 7.28e-16 ***
## grade13 3303587 234361 14.096 < 2e-16 ***
## lat 644972 11166 57.764 < 2e-16 ***
## view 106862 2086 51.234 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 225800 on 21599 degrees of freedom
## Multiple R-squared: 0.6219, Adjusted R-squared: 0.6217
## F-statistic: 2733 on 13 and 21599 DF, p-value: < 2.2e-16mod3b <- lm(price ~ grade + lat + waterfront, data = houses_tidy)
summary(mod3b)##
## Call:
## lm(formula = price ~ grade + lat + waterfront, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1862555 -109244 -24781 70145 4724767
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -30511083 581588 -52.462 < 2e-16 ***
## grade3 190241 262923 0.724 0.469343
## grade4 95796 231583 0.414 0.679130
## grade5 112654 228161 0.494 0.621488
## grade6 153414 227746 0.674 0.500562
## grade7 243828 227703 1.071 0.284264
## grade8 372609 227709 1.636 0.101783
## grade9 594340 227734 2.610 0.009066 **
## grade10 870026 227792 3.819 0.000134 ***
## grade11 1267641 227979 5.560 2.72e-08 ***
## grade12 1902270 228964 8.308 < 2e-16 ***
## grade13 3500877 236288 14.816 < 2e-16 ***
## lat 644910 11259 57.278 < 2e-16 ***
## waterfrontTRUE 851219 18046 47.168 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 227700 on 21599 degrees of freedom
## Multiple R-squared: 0.6156, Adjusted R-squared: 0.6154
## F-statistic: 2661 on 13 and 21599 DF, p-value: < 2.2e-16# view model is best, keep mod3a
par(mfrow = c(2, 2))
plot(mod3a)## Warning: not plotting observations with leverage one:
## 19453
## Warning: not plotting observations with leverage one:
## 19453
houses_resid <- houses_tidy %>%
add_residuals(mod3a) %>%
select(-c("price", "grade", "lat", "view"))
houses_resid_numeric <- houses_resid %>%
select_if(is.numeric)
houses_resid_nonnumeric <- houses_resid %>%
select_if(function(x) !is.numeric(x))
houses_resid_nonnumeric$resid <- houses_resid$residggpairs(houses_resid_numeric)
ggpairs(houses_resid_nonnumeric)
sqft_basement has highest correlation with residuals. Let’s test against all remaining categorical predictors:
mod4a <- lm(price ~ grade + lat + view + sqft_basement, data = houses_tidy)
summary(mod4a)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1569582 -105308 -18663 70867 5230039
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.875e+07 5.647e+05 -50.913 < 2e-16 ***
## grade3 1.830e+05 2.542e+05 0.720 0.47164
## grade4 8.134e+04 2.239e+05 0.363 0.71638
## grade5 1.088e+05 2.206e+05 0.493 0.62197
## grade6 1.355e+05 2.202e+05 0.615 0.53841
## grade7 2.042e+05 2.201e+05 0.927 0.35373
## grade8 3.194e+05 2.201e+05 1.451 0.14687
## grade9 5.281e+05 2.202e+05 2.399 0.01646 *
## grade10 7.859e+05 2.202e+05 3.568 0.00036 ***
## grade11 1.156e+06 2.204e+05 5.244 1.59e-07 ***
## grade12 1.765e+06 2.214e+05 7.974 1.61e-15 ***
## grade13 3.169e+06 2.285e+05 13.869 < 2e-16 ***
## lat 6.079e+05 1.094e+04 55.559 < 2e-16 ***
## view 8.903e+04 2.101e+03 42.367 < 2e-16 ***
## sqft_basement 1.204e+02 3.582e+00 33.600 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 220100 on 21598 degrees of freedom
## Multiple R-squared: 0.6407, Adjusted R-squared: 0.6405
## F-statistic: 2751 on 14 and 21598 DF, p-value: < 2.2e-16mod4b <- lm(price ~ grade + lat + view + waterfront, data = houses_tidy)
summary(mod4b)##
## Call:
## lm(formula = price ~ grade + lat + view + waterfront, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1713543 -105538 -20871 69842 4901827
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -30758475 564698 -54.469 < 2e-16 ***
## grade3 191262 255269 0.749 0.453709
## grade4 84892 224841 0.378 0.705758
## grade5 106294 221518 0.480 0.631345
## grade6 147526 221116 0.667 0.504659
## grade7 236375 221074 1.069 0.284985
## grade8 354372 221081 1.603 0.108970
## grade9 562139 221106 2.542 0.011016 *
## grade10 823728 221164 3.725 0.000196 ***
## grade11 1197915 221350 5.412 6.30e-08 ***
## grade12 1804315 222314 8.116 5.07e-16 ***
## grade13 3351868 229446 14.609 < 2e-16 ***
## lat 650115 10932 59.466 < 2e-16 ***
## view 80422 2217 36.273 < 2e-16 ***
## waterfrontTRUE 582422 19024 30.615 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 221100 on 21598 degrees of freedom
## Multiple R-squared: 0.6377, Adjusted R-squared: 0.6374
## F-statistic: 2715 on 14 and 21598 DF, p-value: < 2.2e-16mod4c <- lm(price ~ grade + lat + view + condition, data = houses_tidy)
summary(mod4c)##
## Call:
## lm(formula = price ~ grade + lat + view + condition, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1665132 -105472 -18392 71578 5447648
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.043e+07 5.673e+05 -53.640 < 2e-16 ***
## grade3 1.472e+05 2.595e+05 0.567 0.570486
## grade4 6.481e+04 2.291e+05 0.283 0.777293
## grade5 7.656e+04 2.258e+05 0.339 0.734605
## grade6 1.165e+05 2.257e+05 0.516 0.605628
## grade7 2.090e+05 2.257e+05 0.926 0.354406
## grade8 3.359e+05 2.257e+05 1.488 0.136720
## grade9 5.476e+05 2.257e+05 2.426 0.015270 *
## grade10 8.154e+05 2.258e+05 3.612 0.000305 ***
## grade11 1.199e+06 2.260e+05 5.307 1.12e-07 ***
## grade12 1.834e+06 2.269e+05 8.082 6.69e-16 ***
## grade13 3.310e+06 2.339e+05 14.148 < 2e-16 ***
## lat 6.432e+05 1.098e+04 58.552 < 2e-16 ***
## view 1.017e+05 2.058e+03 49.425 < 2e-16 ***
## condition2 -4.275e+02 4.466e+04 -0.010 0.992363
## condition3 -6.453e+03 4.154e+04 -0.155 0.876556
## condition4 5.758e+04 4.158e+04 1.385 0.166054
## condition5 1.349e+05 4.180e+04 3.226 0.001258 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 221800 on 21595 degrees of freedom
## Multiple R-squared: 0.6352, Adjusted R-squared: 0.635
## F-statistic: 2212 on 17 and 21595 DF, p-value: < 2.2e-16mod4d <- lm(price ~ grade + lat + view + renovated, data = houses_tidy)
summary(mod4d)##
## Call:
## lm(formula = price ~ grade + lat + view + renovated, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1637628 -105101 -19410 70577 5280904
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -30179084 571198 -52.835 < 2e-16 ***
## grade3 188870 258132 0.732 0.464372
## grade4 75932 227363 0.334 0.738407
## grade5 108602 224003 0.485 0.627808
## grade6 141362 223596 0.632 0.527250
## grade7 229992 223554 1.029 0.303586
## grade8 346281 223561 1.549 0.121413
## grade9 550447 223587 2.462 0.013828 *
## grade10 817501 223645 3.655 0.000257 ***
## grade11 1199432 223833 5.359 8.47e-08 ***
## grade12 1835525 224806 8.165 3.39e-16 ***
## grade13 3275946 232018 14.119 < 2e-16 ***
## lat 637925 11059 57.684 < 2e-16 ***
## view 102285 2076 49.261 < 2e-16 ***
## renovatedTRUE 159554 7606 20.979 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 223500 on 21598 degrees of freedom
## Multiple R-squared: 0.6295, Adjusted R-squared: 0.6292
## F-statistic: 2621 on 14 and 21598 DF, p-value: < 2.2e-16# looks like model with sqft_basement is best, keep mod4a
par(mfrow = c(2, 2))
plot(mod4a)## Warning: not plotting observations with leverage one:
## 19453
## Warning: not plotting observations with leverage one:
## 19453
houses_resid <- houses_tidy %>%
add_residuals(mod4a) %>%
select(- price)Our final model in terms of main effects is: price ~ grade + lat + view + sqft_basement
5 Extensions
Consider possible interactions between your four main effect predictors and test their effect upon \(r^2\). Choose your best candidate interaction and visualise its effect.
Calculate the relative importance of predictors from your best \(4\)-predictor model (i.e. the model without an interaction). Which predictor affects
pricemost strongly?
Solution
Now, for interactions, have six possibilities that obey principle of strong hierarchy (i.e. consider including an interaction only if its main effects are already present in the model)
mod5a <- lm(price ~ grade + lat + view + sqft_basement + grade:lat, data = houses_tidy)
summary(mod5a)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + grade:lat,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1496592 -101302 -20837 67958 5196329
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.375e+07 4.424e+07 -0.989 0.3228
## grade3 3.068e+07 6.120e+07 0.501 0.6162
## grade4 3.760e+07 4.798e+07 0.784 0.4332
## grade5 2.917e+07 4.457e+07 0.654 0.5129
## grade6 2.420e+07 4.428e+07 0.546 0.5848
## grade7 1.912e+07 4.425e+07 0.432 0.6656
## grade8 1.546e+07 4.425e+07 0.349 0.7267
## grade9 -7.530e+05 4.427e+07 -0.017 0.9864
## grade10 -1.492e+07 4.435e+07 -0.336 0.7366
## grade11 -1.469e+07 4.466e+07 -0.329 0.7422
## grade12 -8.399e+07 4.632e+07 -1.813 0.0698 .
## grade13 3.135e+06 2.463e+05 12.728 <2e-16 ***
## lat 9.234e+05 9.308e+05 0.992 0.3212
## view 8.949e+04 2.086e+03 42.892 <2e-16 ***
## sqft_basement 1.208e+02 3.556e+00 33.968 <2e-16 ***
## grade3:lat -6.429e+05 1.290e+06 -0.498 0.6183
## grade4:lat -7.898e+05 1.010e+06 -0.782 0.4340
## grade5:lat -6.115e+05 9.378e+05 -0.652 0.5143
## grade6:lat -5.062e+05 9.315e+05 -0.543 0.5869
## grade7:lat -3.980e+05 9.310e+05 -0.428 0.6690
## grade8:lat -3.186e+05 9.310e+05 -0.342 0.7322
## grade9:lat 2.662e+04 9.315e+05 0.029 0.9772
## grade10:lat 3.296e+05 9.330e+05 0.353 0.7239
## grade11:lat 3.325e+05 9.395e+05 0.354 0.7234
## grade12:lat 1.801e+06 9.745e+05 1.848 0.0646 .
## grade13:lat NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 218400 on 21588 degrees of freedom
## Multiple R-squared: 0.6464, Adjusted R-squared: 0.646
## F-statistic: 1644 on 24 and 21588 DF, p-value: < 2.2e-16mod5b <- lm(price ~ grade + lat + view + sqft_basement + grade:view, data = houses_tidy)
summary(mod5b)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + grade:view,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1878838 -104533 -20264 69483 5165978
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.871e+07 5.606e+05 -51.208 < 2e-16 ***
## grade3 1.828e+05 2.523e+05 0.725 0.468734
## grade4 8.617e+04 2.226e+05 0.387 0.698645
## grade5 1.122e+05 2.189e+05 0.513 0.608291
## grade6 1.379e+05 2.185e+05 0.631 0.528067
## grade7 2.081e+05 2.185e+05 0.952 0.340866
## grade8 3.222e+05 2.185e+05 1.475 0.140313
## grade9 5.373e+05 2.185e+05 2.459 0.013953 *
## grade10 7.518e+05 2.186e+05 3.439 0.000585 ***
## grade11 1.084e+06 2.189e+05 4.953 7.35e-07 ***
## grade12 1.741e+06 2.208e+05 7.887 3.23e-15 ***
## grade13 2.594e+06 2.372e+05 10.939 < 2e-16 ***
## lat 6.070e+05 1.086e+04 55.879 < 2e-16 ***
## view 4.024e+05 3.764e+04 10.689 < 2e-16 ***
## sqft_basement 1.178e+02 3.570e+00 32.987 < 2e-16 ***
## grade3:view NA NA NA NA
## grade4:view -3.485e+05 1.009e+05 -3.454 0.000553 ***
## grade5:view -3.381e+05 4.384e+04 -7.712 1.29e-14 ***
## grade6:view -3.360e+05 3.893e+04 -8.631 < 2e-16 ***
## grade7:view -3.454e+05 3.794e+04 -9.104 < 2e-16 ***
## grade8:view -3.214e+05 3.782e+04 -8.498 < 2e-16 ***
## grade9:view -3.326e+05 3.788e+04 -8.782 < 2e-16 ***
## grade10:view -2.595e+05 3.800e+04 -6.828 8.81e-12 ***
## grade11:view -2.424e+05 3.843e+04 -6.307 2.89e-10 ***
## grade12:view -2.972e+05 4.002e+04 -7.428 1.14e-13 ***
## grade13:view NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 218500 on 21589 degrees of freedom
## Multiple R-squared: 0.6462, Adjusted R-squared: 0.6459
## F-statistic: 1715 on 23 and 21589 DF, p-value: < 2.2e-16mod5c <- lm(price ~ grade + lat + view + sqft_basement + grade:sqft_basement, data = houses_tidy)
summary(mod5c)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + grade:sqft_basement,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2093544 -102714 -19903 68417 4922552
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.922e+07 5.576e+05 -52.398 < 2e-16 ***
## grade3 1.849e+05 2.507e+05 0.737 0.460910
## grade4 7.737e+04 2.210e+05 0.350 0.726274
## grade5 1.098e+05 2.176e+05 0.504 0.613935
## grade6 1.427e+05 2.172e+05 0.657 0.511185
## grade7 2.183e+05 2.172e+05 1.005 0.314862
## grade8 3.305e+05 2.172e+05 1.522 0.128016
## grade9 5.195e+05 2.172e+05 2.392 0.016782 *
## grade10 7.595e+05 2.173e+05 3.495 0.000475 ***
## grade11 1.067e+06 2.176e+05 4.906 9.38e-07 ***
## grade12 1.603e+06 2.191e+05 7.318 2.61e-13 ***
## grade13 2.047e+06 2.372e+05 8.629 < 2e-16 ***
## lat 6.177e+05 1.080e+04 57.166 < 2e-16 ***
## view 8.401e+04 2.100e+03 40.011 < 2e-16 ***
## sqft_basement 9.153e+02 5.204e+01 17.589 < 2e-16 ***
## grade3:sqft_basement NA NA NA NA
## grade4:sqft_basement -6.667e+01 1.106e+03 -0.060 0.951943
## grade5:sqft_basement -7.957e+02 1.065e+02 -7.470 8.33e-14 ***
## grade6:sqft_basement -8.503e+02 5.511e+01 -15.429 < 2e-16 ***
## grade7:sqft_basement -8.442e+02 5.238e+01 -16.118 < 2e-16 ***
## grade8:sqft_basement -8.273e+02 5.241e+01 -15.785 < 2e-16 ***
## grade9:sqft_basement -7.618e+02 5.272e+01 -14.451 < 2e-16 ***
## grade10:sqft_basement -7.237e+02 5.317e+01 -13.611 < 2e-16 ***
## grade11:sqft_basement -6.234e+02 5.421e+01 -11.499 < 2e-16 ***
## grade12:sqft_basement -5.887e+02 5.646e+01 -10.427 < 2e-16 ***
## grade13:sqft_basement NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 217100 on 21589 degrees of freedom
## Multiple R-squared: 0.6505, Adjusted R-squared: 0.6502
## F-statistic: 1747 on 23 and 21589 DF, p-value: < 2.2e-16mod5d <- lm(price ~ grade + lat + view + sqft_basement + lat:view, data = houses_tidy)
summary(mod5d)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + lat:view,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1544431 -103361 -20451 68950 5201833
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.680e+07 5.780e+05 -46.362 < 2e-16 ***
## grade3 1.749e+05 2.530e+05 0.691 0.489328
## grade4 7.915e+04 2.228e+05 0.355 0.722429
## grade5 1.100e+05 2.195e+05 0.501 0.616167
## grade6 1.372e+05 2.191e+05 0.626 0.531099
## grade7 2.062e+05 2.191e+05 0.941 0.346529
## grade8 3.209e+05 2.191e+05 1.464 0.143085
## grade9 5.296e+05 2.191e+05 2.417 0.015660 *
## grade10 7.858e+05 2.192e+05 3.585 0.000338 ***
## grade11 1.152e+06 2.194e+05 5.251 1.53e-07 ***
## grade12 1.757e+06 2.203e+05 7.974 1.61e-15 ***
## grade13 3.133e+06 2.274e+05 13.776 < 2e-16 ***
## lat 5.668e+05 1.125e+04 50.366 < 2e-16 ***
## view -1.069e+07 7.448e+05 -14.352 < 2e-16 ***
## sqft_basement 1.188e+02 3.567e+00 33.320 < 2e-16 ***
## lat:view 2.266e+05 1.566e+04 14.471 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 219100 on 21597 degrees of freedom
## Multiple R-squared: 0.6442, Adjusted R-squared: 0.6439
## F-statistic: 2607 on 15 and 21597 DF, p-value: < 2.2e-16mod5e <- lm(price ~ grade + lat + view + sqft_basement + lat:sqft_basement, data = houses_tidy)
summary(mod5e)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + lat:sqft_basement,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1550147 -101737 -18763 68303 5174888
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.488e+07 6.359e+05 -39.124 < 2e-16 ***
## grade3 1.670e+05 2.532e+05 0.659 0.509600
## grade4 7.862e+04 2.230e+05 0.353 0.724442
## grade5 1.072e+05 2.197e+05 0.488 0.625492
## grade6 1.362e+05 2.193e+05 0.621 0.534705
## grade7 2.060e+05 2.193e+05 0.939 0.347638
## grade8 3.210e+05 2.193e+05 1.464 0.143228
## grade9 5.299e+05 2.193e+05 2.416 0.015700 *
## grade10 7.885e+05 2.194e+05 3.594 0.000326 ***
## grade11 1.158e+06 2.196e+05 5.274 1.35e-07 ***
## grade12 1.765e+06 2.205e+05 8.004 1.26e-15 ***
## grade13 3.143e+06 2.276e+05 13.809 < 2e-16 ***
## lat 5.264e+05 1.256e+04 41.918 < 2e-16 ***
## view 8.911e+04 2.093e+03 42.569 < 2e-16 ***
## sqft_basement -1.698e+04 1.310e+03 -12.968 < 2e-16 ***
## lat:sqft_basement 3.595e+02 2.752e+01 13.060 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 219300 on 21597 degrees of freedom
## Multiple R-squared: 0.6435, Adjusted R-squared: 0.6433
## F-statistic: 2599 on 15 and 21597 DF, p-value: < 2.2e-16mod5f <- lm(price ~ grade + lat + view + sqft_basement + view:sqft_basement, data = houses_tidy)
summary(mod5f)##
## Call:
## lm(formula = price ~ grade + lat + view + sqft_basement + view:sqft_basement,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1638784 -104221 -19570 69709 5166209
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.883e+07 5.636e+05 -51.155 < 2e-16 ***
## grade3 1.833e+05 2.537e+05 0.723 0.469931
## grade4 8.399e+04 2.234e+05 0.376 0.707022
## grade5 1.118e+05 2.201e+05 0.508 0.611432
## grade6 1.385e+05 2.197e+05 0.630 0.528579
## grade7 2.087e+05 2.197e+05 0.950 0.342139
## grade8 3.238e+05 2.197e+05 1.474 0.140599
## grade9 5.312e+05 2.197e+05 2.417 0.015645 *
## grade10 7.884e+05 2.198e+05 3.587 0.000335 ***
## grade11 1.155e+06 2.200e+05 5.250 1.54e-07 ***
## grade12 1.742e+06 2.210e+05 7.886 3.27e-15 ***
## grade13 3.138e+06 2.281e+05 13.756 < 2e-16 ***
## lat 6.096e+05 1.092e+04 55.822 < 2e-16 ***
## view 7.106e+04 2.847e+03 24.962 < 2e-16 ***
## sqft_basement 1.056e+02 3.910e+00 27.003 < 2e-16 ***
## view:sqft_basement 2.870e+01 3.073e+00 9.339 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 219700 on 21597 degrees of freedom
## Multiple R-squared: 0.6422, Adjusted R-squared: 0.6419
## F-statistic: 2584 on 15 and 21597 DF, p-value: < 2.2e-16# mod5c looks like the best
par(mfrow = c(2,2))
plot(mod5c)## Warning: not plotting observations with leverage one:
## 8620, 19453
## Warning: not plotting observations with leverage one:
## 8620, 19453
It seems that the grade:sqft_basement interaction leads to highest \(r^2\) (but two levels of the interaction cannot be determined due to fitting problems).
Now let’s see a visualisation of the effect of this interaction.
houses_resid %>%
ggplot(aes(x = sqft_basement, y = resid, colour = grade)) +
geom_point(alpha = 0.5) +
geom_smooth(method = "lm", se = FALSE) +
facet_wrap(~ grade)## `geom_smooth()` using formula 'y ~ x'
Relative importance of predictors:
library(relaimpo)## Loading required package: MASS##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select## Loading required package: boot## Loading required package: survey## Loading required package: grid## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpack## Loading required package: survival##
## Attaching package: 'survival'## The following object is masked from 'package:boot':
##
## aml##
## Attaching package: 'survey'## The following object is masked from 'package:graphics':
##
## dotchart## Loading required package: mitools## This is the global version of package relaimpo.## If you are a non-US user, a version with the interesting additional metric pmvd is available## from Ulrike Groempings web site at prof.beuth-hochschule.de/groemping.calc.relimp(mod4a, method = "lmg", rela = TRUE)## Response variable: price
## Total response variance: 134782378397
## Analysis based on 21613 observations
##
## 14 Regressors:
## Some regressors combined in groups:
## Group grade : grade3 grade4 grade5 grade6 grade7 grade8 grade9 grade10 grade11 grade12 grade13
##
## Relative importance of 4 (groups of) regressors assessed:
## grade lat view sqft_basement
##
## Proportion of variance explained by model: 64.07%
## Metrics are normalized to sum to 100% (rela=TRUE).
##
## Relative importance metrics:
##
## lmg
## grade 0.67013680
## lat 0.10999806
## view 0.13594911
## sqft_basement 0.08391603
##
## Average coefficients for different model sizes:
##
## 1group 2groups 3groups 4groups
## grade3 63666.6672 105085.5034 144251.4794 182970.8245
## grade4 72381.0351 74840.3850 77838.0192 81340.5229
## grade5 106523.9716 106120.0182 106860.8796 108762.8919
## grade6 159919.6380 148997.5902 140821.1511 135460.9340
## grade7 260590.2629 235879.0272 216962.5200 204157.6486
## grade8 400852.7662 366263.3497 339139.6632 319376.1318
## grade9 631513.1864 588676.4480 554462.1758 528149.5236
## grade10 929771.0746 870400.5336 822843.4234 785879.9257
## grade11 1354841.7274 1272639.8932 1206980.5470 1155890.1732
## grade12 2049222.0006 1930531.4478 1836851.8205 1765372.6480
## grade13 3567615.3852 3398156.8184 3266023.2283 3169155.3827
## lat 813411.5832 722508.3996 660525.2637 607867.6492
## view 190335.2479 151137.4701 117930.1234 89031.6774
## sqft_basement 268.6136 203.8056 154.5114 120.3599It looks like the grade of property is the most important determiner of price, followed by the number of views the property has received.